https://pbeltran.github.io/afectoeos-feb2020

Introduction

The OSA has been applied to the epistemic (institutional knowledge) and cognitive (personal or subjective knowledge) domain, taking as primitive notions the following:

• Situation-problem
• Practice (mathematics)
• Object (emerging and intervening in practices)
• Meaning (relation between objects) (Font, et al., 2013).

Introduction

These primary entities can be analysed from different points of view (dualities or contextual polarities), giving rise to new types of secondary entities: personal-institutional, ostensive-non-ostensive, extensive-intensive, unitary-systemic, expression-content.

Our purpose here is to address the following questions: Is an onto-semiotic approach relevant to the study of the affective domain? Is it possible to bring new insights to affect in mathematics education when tackling it with the OSA theoretical lenses? What theoretical models on affect can be incorporated and aligned with this approach?

Affect and mathematics education

Anyway, most researchers accept emotions, attitudes, beliefs and values as the key components of the affective domain in mathematics education and use them to study the interactions among cognition, mathematical affect, teaching and learning processes, problem solving, achievement and behaviour (Grootenboer & Marshman, 2016; Pepin & Roesken-Winter, 2015).

One of these interactions is the mediation of beliefs in the learning itself. Furinghetti and Pehkonen (2002) suggest that beliefs can be considered also as a form of subjective knowledge and thus they can be interpreted as a nexus between the cognitive and the affective domains.

Besides, there are also complex interactions between teacher and student beliefs, particularly in problem-solving approaches and ICT mediation (Depaepe, De Corte, & Verschaffel, 2015; Gómez-Chacón, 2011).

Our literature review suggests that there is a clear interest in deepening how the different facets of the affective domain interact among them and with other domains, for what an onto-semiotic approach can be useful, as it incorporates theoretical notions to handle the multiple facets of a mathematics learning and teaching process.

Didactical fact

A didactical fact is ‘any event that has a place and a time in the becoming of a mathematical instruction process and that, for some reason, is considered as a unit’ (Wilhelmi, Font, &Godino, 2005). We will say that it is a meaningful didactic fact (MDF) ‘if the didactic actions or practices that compose it play a role, or admit an interpretation, in terms of the intended instructional objective’ (Godino, Rivas, Arteaga, Lasa, & Wilhelmi, 2014).

To support the description and understanding of the categories of affective entities, we will apply the notion of MDF to two experiences made in different educational and research contexts.

The selected MDF from both data sources allow us to illustrate specific aspects of the affective domain, as we will see below, through the OSA categories.

Affective meaning of certain signs

Following the OSA pragmatic epistemological assumptions, we are now asking for the affective meaning of certain signs (in the sense of Peirce’s representamen), in any of the possible registers and representations, which may be verbal or written expressions, observable behaviours, etc. Such meaning must be sought in the systems of practices that a person performs to solve a problem situation, or towards a practice, an object, a mathematical process, or any mathematics study situation.

Affective domain components

There is agreement, within the scope of research in mathematical education, that the affective domain consists of three components: emotions, attitudes and beliefs. The origins of this classification go back to McLeod (1992) and, in this article, we will use this ontology of affective objects, to which we will add the values, construct included in the model of DeBellis & Goldin (2006).

Affective situations

It is necessary to consider a specific type of situation that provides the appropriate framework for describing affective practices. When a student is confronted with a situation-problem, an affective situation occurs that juxtaposes itself with the cognitive one, and which comes to include the purely personal meanings about it, in the form of emotions, attitudes, beliefs or values.

However, affective situations do not arise solely in response to a problem-situation, since the teaching and learning ecosystems provide constant reference points for the affective domain. In this way, there are situations of production, communication or, simply, of individual mathematical study.

The teacher may pose situations in which, specifically, the students’ beliefs are brought into play towards a concrete mathematical object.

Affective practices

Affective practices are any action or affective manifestation that accompanies any mathematical practice. They can be manifestations about emotions, attitudes, beliefs or values about the objects put into play. Each of these affect expressions can vary in intensity throughout a practice or even disappear, giving rise to new manifestations. The great part of the affective trajectory remains hidden from the eyes of the teacher, because not all the affective states are manifested. Besides, it is not possible for a single person to observe the whole group to interpret small gestures or signs of every student.

Nevertheless, an observation record, as a classroom diary (Porlán and Martín, 1991), helps to collect data on which to reflect later. And, in addition, there are instruments that can be incorporated into the teaching practice to gather information about the affective domain.

Intervening and emerging objects

Although the categorisation of the affective domain in emotions, attitudes and beliefs is accepted by the research community, to which values can be added, the meaning of such constructs is still a matter of controversy: - Emotions: quickly changing feelings experienced in a conscious way or occurring pre-consciously or unconsciously during mathematical (or other) activity. Emotions vary from mild to intense and are locally and contextually immersed. - Attitudes: describe orientations or predispositions towards certain sets of emotional sensations (positive or negative), in particular (mathematical) contexts. This differs from the more common view of attitudes as predispositions toward certain patterns of behaviour. Attitudes are moderately stable, implying an interactive balance between affection and cognition.

• Beliefs: they imply the attribution of some kind of truth or external validity to the system of propositions or other cognitive configurations. Beliefs are often highly stable, largely cognitive and structured, in which emotions and attitudes intersect with them, contributing to their stabilisation.
• Values: including ethical and moral components, refers to personal truths or commitments deeply appreciated by individuals. They help to motivate long-term decisions or set short-term priorities. They can be highly structured, building value systems.

Given the interaction with the cognitive domain, it may be convenient to consider, as a category of affective objects, the various modes of expression of the affects: gestures, terms of ordinary language, etc. (Álvarez, 2012), which would constitute the ostensible facet of affections. Emotions, attitudes, beliefs and values are relative to mathematical situations and practices, and to the distinct primary mathematical objects. It makes sense, therefore, to research the affective components towards the demonstrations, the procedures, the representations, etc. Figure 1 summarises the primary affective-cognitive categories.

The characteristics of affective languages, which could be considered as a fifth category of affective objects, expand the semiotic registers and representations that emerge from the practices, since much of the affective charge is expressed nonverbally, within a system of information transmission, in which each element is interpreted by the different agents involved (teacher, students).

Emotions, therefore, can arise as an instantaneous emotional response to a sensorial stimulus, which may have a mathematical character (a field of problems) or not (going to school). Although this distinction seems trivial, the origin of emotions is complex to interpret. Consider the MDF3, in which the annotation in the teacher’s diary indicates that his students show nervousness and agitation because of the proximity of a written test: MDF3: Nervousness because of the proximity of a written evaluation test. [Teacher’s diary] I tell the students that they already know the subject and that the examination date will be in 7 days. They complain, arguing that they have another test that day. Most of them, who were quite distracted and talking about other things, join in the discussion. Whereas it is clear that this is an emotional response, instantaneous, to a particular stimulus (the teacher announcing the examination date), the actual origin of the emotion could be based on specific beliefs about math tests or more general beliefs about school. Other times, it is easier to identify the source of those emotions, although it is not possible to establish absolute certainty, which would require more data collection tools. For example, the MDF4 describes an emotional reaction that could be named curiosity, to comments of the teacher that try to confront the students’ beliefs about random experiments with a specific mathematical fact, in this case, the stability of the relative frequencies. Some students hold this emotion, which leads to an attitude of interest: MDF4: Emotion that arouses interest. [Teacher’s diary]. I briefly introduce the stability of the relative frequencies, summarising the previous day activity (coin tosses). I ask them about the results of the coins, ‘to what number the percentage comes up’, to which they say that ‘one goes up and the other one goes down’. I see that at least A7 looks interested. I suggest them to think about what would happen if the coins were tossed 1000 times. The feedback system formed by the different components of the affective domain is put into play in any type of situation. The MDF5 shows how, when facing a problem-situation, some students show mental block emotion, while others start from a passive attitude, which they have been able to reach from the previous emotion or from causes unrelated to the situation. One of the teacher objectives in these cases is to intervene in the feedback loop (attitude-emotional) to promote progress in the teaching-learning process: MDF5: Emotion and attitude within a situation-problem. [Teacher’s diary] I see that there are students who are finishing, but also some who go slowly, either because they do not want to do it (case of A6) or because they get stuck. In the case of A6, I urge her to finish it. The students to whom the teaching-learning sequences are directed present beliefs about the mathematical objects that make up the trajectory of the instructional process. In the case of probability, its different meanings (Batanero, 2005) must be negotiated from personal belief systems, as seen in MDF6, where students, used to solve similar problem- situations with other procedures, are reluctant to use a new one: MDF6: Belief about the procedure to solve a situation-problem. [Teacher’s diary] I see that some of them have already reached an exercise in which they get stuck, and I introduce the tree diagrams, as a helpful way to solve it. But I notice that some students are still trying to do it without using the diagram and without success. At other times, in a dialogic interaction environment, it is the teacher who decides to inquire directly about the students’ beliefs. The MDF7 shows an example of this, in which the teacher asks about the distinction between random and deterministic phenomena, an issue that relates to the perception of chance: MDF7: Beliefs about random phenomena. [Teacher’s diary] They have no problem to specify the sample space of these experiments (balls extraction, pushpin that is thrown to the ground) or to distinguish if they are random or deterministic. I take the time to ask if predicting tomorrow weather will be random or deterministic. Ethical and moral values differ from beliefs in which, while the latter constitute judgments of subjective truth from the logical or empirical point of view, values refer to purely personal choices (that which is good, or desirable) (Goldin, 2002). However, belief systems and value systems are closely related, and at times, it is difficult and inoperative, to isolate them. The MDF8 exemplifies how the commitment to the study process (a personal choice that constitutes a value) influences the learning trajectory, directly reducing the effective teaching time: MDF8: Value about commitment to the study process. [Teacher’s diary] They take a long time to come from recess. I must raise my voice and show myself authoritative, so they can take out a notebook and a book. A5 takes even longer, speaks and laughs with his mates and has not brought the book. On the other hand, affective languages deserve special attention, and this is reflected in the key place reserved for them in Figure 1. Language, in its different registers, constitutes not only a communicative vehicle, but, being formed by signs which are constantly interpreted, is a tool of signification. In the case of the affective domain, non-verbal communication plays a fundamental role (Johnson, 1999; Knapp, Hall, & Horgan, 2013). In the same way that pupils’ productions, both written (also in their different registers) and verbal, provide indicators about the cognitive domain, the transmission of much of the affective information is done through facial expressions, gestures, postures, movements, etc. Harris and Rosenthal’s (2005) meta-study shows how students improve in certain facets when the teacher’s non-verbal language includes immediacy signs, such as gesturing when speaking, not sitting behind his desk, looking at students while talking, smiling, using a tone that is not monotonous, etc. Thus, students show interest in the course and the teacher, pay attention and have the perception that they have learned a lot in class (Rocca, 2004). Likewise, the results of his study also show correlations between the teacher’s non-verbal language and students’ cognitive performance, although this is something that is under study (Witt, Wheeless, & Allen, 2004). All these affective languages match interaction patterns that can be encompassed into one of the following three dimensions (Rompelman, 2002): opportunity to respond in a climate of trust, possibility of feedback, and consideration towards people (respect). Harris and Rosenthal (2005) also point out the difficulty of investigating empirically in the classroom environment, due to the apparatus required to capture all non-verbal information. Other authors agree in this regard. Mitchell (2013) points out that, given the positive relationship between the teacher’s non-verbal language and student attitudes, it is important that the teacher is not only enthusiastic about content, but should also show that enthusiasm to have a positive impact in the learning of the students. This influence of the teacher in the students’ beliefs towards mathematics and that, in the end, influence the other affective components, is evident in the essay excerpt shown here: Mathematics was never my strong point, rather, my Achilles heel. I came to hate them when I was in High School, despite this, little by little I can understand them thanks to my perseverance and dedication. Depending on the syllabus, I pay more attention when I find it interesting, although if I get bored I disconnect. Also depending on the teacher, which plays a fundamental role in making learning easier. In the excerpt, the student mentions the importance of certain attitudes (constancy, perseverance) and of the emotions that are awakened by some content (boredom), as well as the role that the teacher plays to encourage interest.

The affective facet is perhaps the most complex to evaluate within the six facets that make up the didactical suitability in the OSA, and an initial proposal was suggested in Godino (2013). Didactical suitability is understood as the degree to which an instructional process (or a part of it) combines certain characteristics in order to be classified as suitable (optimal or appropriate) for the adaptation between the personal meanings obtained by students (learning), and the intended or implemented institutional meanings (teaching), taking into consideration the circumstances and the available resources (environment). The didactical suitability notion is broken down into six specific facets (epistemic, cognitive, affective, interactional, mediational and ecological suitability) (Breda, Pino-Fan, & Font, 2017). This notion is being widely used in research on teachers’ education as a tool that supports teacher’s reflection on their own practice (Breda, Font, & Pino-Fan, 2018).

The literature review clearly indicates that the relationship with the students’ perception of their own learning influences their own progress. There is also a certain correlation between affection and degree of cognitive performance, although more studies are needed in this line (Gómez-Chacón, 2017).

This work constitutes an advance within the OSA theoretical framework, when applying their primary categories and their ontology to the affective domain of the teaching-learning processes. The pertinence of the study is reflected both in the articulating nature between representational systems of theoretical models and in the revision of the original criteria on affective suitability proposed by Godino (2013).

Referencias

Las referencias mencionadas pueden encontrarse en el artículo completo:

Beltrán-Pellicer, P., & Godino, J. D. (2020). An onto-semiotic approach to the analysis of the affective domain in mathematics education. Cambridge Journal of Education, 50(1), 1-20. DOI: 10.1080/0305764X.2019.1623175

Créditos

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